We introduce the concept of fundamental sequence for a finite graded poset
The study of the order relation in particular subsets of integer partitions has been recently addressed from the point of view of the discrete dynamical models. The excellent survey [
One grain can move from a column to the next one if the difference of height of these two columns is greater than or equal to 2.
In the model
If a column containing
There are a lot of specializations and extensions of this model which have been introduced and studied under different names, different aspects, and different approaches. The
There exists a wide literature concerning dynamical models related in several ways to the original model
When we study the parallel dynamic of an SPM which also has a lattice structure, starting from its initial configuration (denoted by
If
Parallel rank in
Case | Relation between |
Parallel rank |
---|---|---|
I |
|
|
II |
|
|
III |
|
|
IV |
|
|
V |
|
|
VI |
|
|
VII |
|
|
VIII |
|
|
Fundamental sequence in
Case | Fundamental sequence |
---|---|
Case I |
|
| |
Case II |
|
| |
Case III |
|
| |
Case IV |
|
| |
Case V |
|
| |
Case VI.1 |
|
| |
Case VI.2 |
|
| |
Case VI.3 |
|
| |
Case VI.4 |
|
| |
Case VI.5 |
|
| |
Case VI.6 |
|
| |
Case VII.1 |
|
| |
Case VII.2 |
|
| |
Case VIII.1 |
|
| |
Case VIII.2 |
|
| |
Case VIII.3 |
|
In this section, we introduce the lattice of signed integer partitions
In some cases, when
Now let us describe the evolution rules which describe
Let
The choice to set
If
One grain must be deleted from the
If
We start to show that
We must show now that
Therefore, if there exists exactly an index there exits exactly an index
According to the definition of
Now let us study the sequential and the parallel dynamic of the model
Let one denote by
Let
For the sake of completeness, we recall that in [
We recall now the concept of involution poset. An if
The map
Now, if
We have:
(i)
(ii)
(i) It is sufficient to observe that
(ii) We define the map
The parallel rank
The fundamental sequence of
By Proposition
In this section, we consider a submodel of
(i) In the previous rule, the lowest not empty plus pile can also be the invisible column in the place
(ii) We take implicitly for intended that the shift of one minus singleton pile into a plus singleton pile can be made if the number of plus piles (excluding
We write
If
We start to show that
We suppose at first that
If
An analogous statement of the previous theorem was proven in [
In this section, we study the parallel dynamic of the model
The map
We denote by
By using the rank function, we can compute the sequential convergence time of any element
In particular, if we denote by
the following result holds:
The minimal and maximal elements in
The next theorem is the main result of this section. Here we determine the parallel rank of the lattice
If If If If If If If If
In all cases, let us denote the sequence of configurations as in (
Since
Now, from
In a similar way, we can compute in all other cases the parallel rank and the fundamental sequence. The details are left to the interested reader. Here are the results.
If If If If If If
In those cases
If If
Therefore, in this case,
If If If
In these cases
We summarize all the previous results in Tables
Let us consider the lattice
In this work, we have introduced a refinement of the concept of parallel time convergence for a finite deterministic discrete dynamical model